Given The Discrete Random Variable $X$, What Is The Probability Distribution Of $X$ If $X \sim B\left(2, \frac{5}{17}\right$\]?A. $\[ \begin{array}{l} P(X=0)=0.415 \\ P(X=1)=0.498

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Understanding the Binomial Distribution of a Discrete Random Variable

In probability theory, a discrete random variable is a variable that can take on a countable number of distinct values. The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, where each trial has a constant probability of success. In this article, we will explore the probability distribution of a discrete random variable XX that follows a binomial distribution with parameters n=2n=2 and p=517p=\frac{5}{17}.

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials. The probability mass function (PMF) of the binomial distribution is given by:

P(X=k)=(nk)pk(1−p)n−kP(X=k) = \binom{n}{k} p^k (1-p)^{n-k}

where nn is the number of trials, kk is the number of successes, pp is the probability of success, and (nk)\binom{n}{k} is the binomial coefficient.

Calculating the Probability Distribution of XX

Given that X∼B(2,517)X \sim B\left(2, \frac{5}{17}\right), we can calculate the probability distribution of XX using the binomial distribution formula. We have n=2n=2 and p=517p=\frac{5}{17}.

Calculating P(X=0)P(X=0)

To calculate P(X=0)P(X=0), we need to plug in k=0k=0 into the binomial distribution formula:

P(X=0)=(20)(517)0(1−517)2−0P(X=0) = \binom{2}{0} \left(\frac{5}{17}\right)^0 \left(1-\frac{5}{17}\right)^{2-0}

Simplifying the expression, we get:

P(X=0)=(1−517)2P(X=0) = \left(1-\frac{5}{17}\right)^2

Evaluating the expression, we get:

P(X=0)=(1217)2=144289≈0.498P(X=0) = \left(\frac{12}{17}\right)^2 = \frac{144}{289} \approx 0.498

Calculating P(X=1)P(X=1)

To calculate P(X=1)P(X=1), we need to plug in k=1k=1 into the binomial distribution formula:

P(X=1)=(21)(517)1(1−517)2−1P(X=1) = \binom{2}{1} \left(\frac{5}{17}\right)^1 \left(1-\frac{5}{17}\right)^{2-1}

Simplifying the expression, we get:

P(X=1)=2(517)(1−517)P(X=1) = 2 \left(\frac{5}{17}\right) \left(1-\frac{5}{17}\right)

Evaluating the expression, we get:

P(X=1)=2(517)(1217)=120289≈0.415P(X=1) = 2 \left(\frac{5}{17}\right) \left(\frac{12}{17}\right) = \frac{120}{289} \approx 0.415

In this article, we have explored the probability distribution of a discrete random variable XX that follows a binomial distribution with parameters n=2n=2 and p=517p=\frac{5}{17}. We have calculated the probability distribution of XX using the binomial distribution formula and found that P(X=0)≈0.498P(X=0) \approx 0.498 and P(X=1)≈0.415P(X=1) \approx 0.415.

  • [1] Johnson, N. L., Kotz, S., & Kemp, A. W. (1992). Univariate discrete distributions. John Wiley & Sons.
  • [2] Ross, S. M. (2010). Introduction to probability models. Academic Press.

The binomial distribution is a fundamental concept in probability theory and has numerous applications in statistics, engineering, and finance. The probability distribution of a discrete random variable XX that follows a binomial distribution can be calculated using the binomial distribution formula. In this article, we have provided a step-by-step guide on how to calculate the probability distribution of XX using the binomial distribution formula.

  • Q: What is the binomial distribution? A: The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, where each trial has a constant probability of success.
  • Q: How do I calculate the probability distribution of a discrete random variable XX that follows a binomial distribution? A: You can calculate the probability distribution of XX using the binomial distribution formula: P(X=k)=(nk)pk(1−p)n−kP(X=k) = \binom{n}{k} p^k (1-p)^{n-k}.
  • Q: What are the parameters of the binomial distribution? A: The parameters of the binomial distribution are nn (the number of trials) and pp (the probability of success).
    Frequently Asked Questions: Binomial Distribution =====================================================

Q: What is the binomial distribution?

A: The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, where each trial has a constant probability of success.

Q: What are the parameters of the binomial distribution?

A: The parameters of the binomial distribution are nn (the number of trials) and pp (the probability of success).

Q: How do I calculate the probability distribution of a discrete random variable XX that follows a binomial distribution?

A: You can calculate the probability distribution of XX using the binomial distribution formula: P(X=k)=(nk)pk(1−p)n−kP(X=k) = \binom{n}{k} p^k (1-p)^{n-k}.

Q: What is the difference between the binomial distribution and the normal distribution?

A: The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, while the normal distribution is a continuous probability distribution that models the behavior of a large number of independent random variables.

Q: When should I use the binomial distribution?

A: You should use the binomial distribution when you are modeling the number of successes in a fixed number of independent trials, where each trial has a constant probability of success.

Q: What are some common applications of the binomial distribution?

A: The binomial distribution has numerous applications in statistics, engineering, and finance, including:

  • Modeling the number of defects in a manufacturing process
  • Modeling the number of successes in a series of independent trials
  • Modeling the behavior of a large number of independent random variables

Q: How do I calculate the mean and variance of a binomial distribution?

A: The mean of a binomial distribution is given by npnp, and the variance is given by np(1−p)np(1-p).

Q: What is the relationship between the binomial distribution and the Poisson distribution?

A: The binomial distribution can be approximated by the Poisson distribution when the number of trials is large and the probability of success is small.

Q: How do I calculate the probability of a binomial distribution using a calculator or computer?

A: You can use a calculator or computer to calculate the probability of a binomial distribution using the binomial distribution formula: P(X=k)=(nk)pk(1−p)n−kP(X=k) = \binom{n}{k} p^k (1-p)^{n-k}.

Q: What are some common mistakes to avoid when working with the binomial distribution?

A: Some common mistakes to avoid when working with the binomial distribution include:

  • Assuming that the binomial distribution is a continuous probability distribution
  • Failing to account for the number of trials and the probability of success
  • Using the binomial distribution to model a continuous random variable

Q: How do I choose the correct binomial distribution parameters?

A: You should choose the correct binomial distribution parameters by considering the number of trials and the probability of success in your problem.

Q: What are some real-world examples of the binomial distribution?

A: Some real-world examples of the binomial distribution include:

  • Modeling the number of defects in a manufacturing process
  • Modeling the number of successes in a series of independent trials
  • Modeling the behavior of a large number of independent random variables

The binomial distribution is a fundamental concept in probability theory and has numerous applications in statistics, engineering, and finance. By understanding the binomial distribution and its parameters, you can model a wide range of real-world phenomena and make informed decisions.